Page 66 - Mechatronics with Experiments

52 MECHATRONICS
Therefore,
∞ ∞
∑ 1 ∑ j( 2 )n⋅t
(t − kT) = e T (2.8)
T
k=−∞ n=−∞
The Laplace transform of the sampled signal is a two-sided Laplace transform, see [5]
∞ ∞
∑ −st
L{y(kT)} = y(t) (t − kt)e dt
∫
−∞ n=−∞
∞
∞ 1 ∑ 2
j
nt −st
= y(t) e T e dt
∫ T
−∞ n=−∞
∞ ∞
1 ∑ −(s−j 2 n)t
= y(t)e T dt (2.9)
T ∫ −∞
n=−∞
The relationship between the Laplace transform of the sampled signal and the Laplace
transform of the original continuous signal is
∞
∑
∗
Y (s) = 1 Y(s − jw n) (2.10)
s
T
n=−∞
where w = 2 is the sampling frequency and T is the sampling period.
s T
Question 2 The Fourier transform of a signal can be obtained from the Laplace
transform by substituting jw in place of s in the Laplace transform of the function. Therefore,
we obtain the following relationship between the Fourier transforms of the sampled and
continuous signal (Figure 2.6),
∞
∑
∗
Y (jw) = 1 Y(jw − jw n) (2.11)
s
T
n=−∞
∗
where Y (jw) is the Fourier transform of sampled signal, and Y(jw) is the Fourier transform
of the original signal.
The frequency content of the sampled signal is the frequency content of the original
signal plus the same content shifted in the frequency axis by integer multiples of the
sampling frequency. In addition, the magnitude of the frequency content is scaled by the
sampling period. The physical interpretation of the above relation is the famous sampling
theorem, also called the Shannon’s sampling theorem.

Sampling Theorem In order to recover the original signal from its samples, the
sampling frequency, w , must be at least two times the highest frequency content, w max ,of
s
the signal,
w ≥ 2 ⋅ w max (2.12)
s
Question 3 We now consider various implications of the sampling operation.

(i) Aliasing: Aliasing is the result of violating the sampling theorem, that is
w < 2 ⋅ w max (2.13)
s
The high frequency components of the original signal show up in the sampled signal
as if they are low frequency components (Figure 2.12). This is called the aliasing.The